A modern Hamiltonian theory offering a unified treatment of all types of systems (i.e. finite, lattice, and field) is presented. Particular attention is paid to nonlinear systems that have more than one Hamiltonian formulation in a single coordinate system. As this property is closely related to integrability, this book presents an algebraic theory of integrable systems. The book is intended for scientists, lecturers, and students interested in the field.
A modern Hamiltonian theory offering a unified treatment of all types of systems (i.e. finite, lattice, and field) is presented. Particular attention is paid to nonlinear systems that have more than one Hamiltonian formulation in a single coordinate system. As this property is closely related to integrability, this book presents an algebraic theory of integrable systems. The book is intended for scientists, lecturers, and students interested in the field.
1. Preliminary Considerations.- 2. Elements of Differential Calculus for Tensor Fields.- 2.1 Tensors.- 2.2 Tensor Fields.- 2.3 Transformation Properties of Tensor Fields.- 2.4 Directional Derivative of Tensor Fields.- 2.5 Differential ?-Forms.- 2.6 Flows and Lie Transport.- 2.7 Lie Derivatives.- 3. The Theory of Hamiltonian and Bi-Hamiltonian Systems.- 3.1 Lie Algebras.- 3.2 Hamiltonian and Bi-Hamiltonian Vector Fields.- 3.3 Symmetries and Conserved Quantities of Dynamical Systems.- 3.4 Tensor Invariants of Dynamical Systems.- 3.5 Algebraic Properties of Tensor Invariants.- 3.6 The Miura Transformation.- 4. Lax Representations of Multi-Hamiltonian Systems.- 4.1 Lax Operators and Their Spectral Deformations.- 4.2 Lax Representations of Isospectral and Nonisospectral Hierarchies.- 4.3 The Lax Operator Algebra.- 5. Soliton Particles.- 5.1 General Aspects.- 5.2 Algebraic Structure of Linear Systems.- 5.3 Algebraic Structure of Multi-Soliton Representation.- 5.4 Multi-Soliton Perturbation Theory.- 6. Multi-Hamiltonian Finite Dimensional Systems.- 6.1 Stationary Flows of Infinite Systems. Ostrogradsky Parametrizations.- 6.2 Stationary Flows of Infinite Systems. Newton Parametrization.- 6.3 Constrained Flows of Lax Equations.- 6.4 Restricted Flows of Infinite Systems.- 6.5 Separability of Bi-Hamiltonian Chains with Degenerate Poisson Structures.- 6.6 Nonstandard Multi-Hamiltonian Structures and Their Finite Dimensional Reductions.- 6.7 Bi-Hamiltonian Chains on Poisson-Nijenhuis Manifolds.- 7. Multi-Hamiltonian Lax Dynamics in (1+1)-Dimensions.- 7.1 Hamiltonian Dynamics on Lie Algebras.- 7.2 Basic Facts About R-Structures.- 7.3 Multi-Hamiltonian Dynamics of Pseudo-Differential Lax Operators.- 7.4 Multi-Hamiltonian Dynamics of Shift Lax Operators.- 8. Towards aMulti-Hamiltonian Theory of (2+1)-Dimensional Field Systems.- 8.1 The Sato Theory.- 8.2 Multi-Hamiltonian Lax Dynamics for Noncommutative Variables.- References.
1. Preliminary Considerations.- 2. Elements of Differential Calculus for Tensor Fields.- 2.1 Tensors.- 2.2 Tensor Fields.- 2.3 Transformation Properties of Tensor Fields.- 2.4 Directional Derivative of Tensor Fields.- 2.5 Differential ?-Forms.- 2.6 Flows and Lie Transport.- 2.7 Lie Derivatives.- 3. The Theory of Hamiltonian and Bi-Hamiltonian Systems.- 3.1 Lie Algebras.- 3.2 Hamiltonian and Bi-Hamiltonian Vector Fields.- 3.3 Symmetries and Conserved Quantities of Dynamical Systems.- 3.4 Tensor Invariants of Dynamical Systems.- 3.5 Algebraic Properties of Tensor Invariants.- 3.6 The Miura Transformation.- 4. Lax Representations of Multi-Hamiltonian Systems.- 4.1 Lax Operators and Their Spectral Deformations.- 4.2 Lax Representations of Isospectral and Nonisospectral Hierarchies.- 4.3 The Lax Operator Algebra.- 5. Soliton Particles.- 5.1 General Aspects.- 5.2 Algebraic Structure of Linear Systems.- 5.3 Algebraic Structure of Multi-Soliton Representation.- 5.4 Multi-Soliton Perturbation Theory.- 6. Multi-Hamiltonian Finite Dimensional Systems.- 6.1 Stationary Flows of Infinite Systems. Ostrogradsky Parametrizations.- 6.2 Stationary Flows of Infinite Systems. Newton Parametrization.- 6.3 Constrained Flows of Lax Equations.- 6.4 Restricted Flows of Infinite Systems.- 6.5 Separability of Bi-Hamiltonian Chains with Degenerate Poisson Structures.- 6.6 Nonstandard Multi-Hamiltonian Structures and Their Finite Dimensional Reductions.- 6.7 Bi-Hamiltonian Chains on Poisson-Nijenhuis Manifolds.- 7. Multi-Hamiltonian Lax Dynamics in (1+1)-Dimensions.- 7.1 Hamiltonian Dynamics on Lie Algebras.- 7.2 Basic Facts About R-Structures.- 7.3 Multi-Hamiltonian Dynamics of Pseudo-Differential Lax Operators.- 7.4 Multi-Hamiltonian Dynamics of Shift Lax Operators.- 8. Towards aMulti-Hamiltonian Theory of (2+1)-Dimensional Field Systems.- 8.1 The Sato Theory.- 8.2 Multi-Hamiltonian Lax Dynamics for Noncommutative Variables.- References.
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